{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Applied Science, Faculty of","@language":"en"},{"@value":"Mechanical Engineering, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Brooks, Peter Noel Hamilton","@language":"en"}],"DateAvailable":[{"@value":"2011-12-08T19:25:44Z","@language":"en"}],"DateIssued":[{"@value":"1960","@language":"en"}],"Degree":[{"@value":"Master of Applied Science - MASc","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"Aerodynamically bluff elastic structures such as suspension bridges and industrial smokestacks have been observed to vibrate violently in the presence of a low speed wind. Although a number of theories such as Den Hartog's quasi-steady instability criterion and the vortex resonance theory have been proposed to explain the phenomenon, the problem is still not completely solved nor yet fully understood. The purpose of this research was to dynamically test a number of two-dimensional cylinders of simple cross-section and to observe whether or not the results correlated with either of the two theories mentioned above. Necessary aerodynamic coefficients were obtained by the graphical integration of measured surface pressure distributions. Tests were performed on models mounted elastically with six degrees of freedom. Generally only one of two modes of vibration was excited at a given airspeed. Dynamic response curves are presented for several lightly damped cylinders. For cylinders with rectangular\r\ncross-section of length\/width (b\/h\/ greater than 0.75, vibration occurred at any airspeed above a certain minimum which depended on the structural damping (galloping).\r\nCylinders with b\/h less than 0.683 were found to vibrate over a limited range of airspeeds which always included the critical velocity for resonance with the periodic formation of vortices in the wakes. Using quasi-steady theory, it was found that the D-section and rectangles with b\/h less than 0.683 have zero aerodynamic damping to large relative angles of attack, due to the symmetry of the wake pressures at angles of attack greater than zero. Because of this the D-section is subject to galloping only when given a substantial initial amplitude. Vortex resonance was observed for all the cylinders tested with two exceptions;\r\nthe reversed D-section and the D-section with the flat face initially at an angle of attack greater than 40\u00b0, both of which appear to be completely stable. Measurements of the frequency of vortex formation for stationary cylinders gave Strouhal numbers which showed only slight variation over the speed range used in the tests, However, a strong variation with b\/h was noted for the rectangles 1.0 < b\/h < 3.0. During vibration of any kind, the frequency of vortex formation was controlled at the frequency of vibration which in all cases was a natural frequency of the system being tested. An energy balance based on quasi-steady theory and neglecting structural damping yields velocity-amplitude curves which give good agreement with the experimental data for the galloping D-section and the square section at various initial angles of attack. The test results indicate that the steady state aerodynamic coefficients provide a useful approximation to the dynamic values; they also indicate that any theory which will completely predict the behaviour of such systems must include the effects of both negative aerodynamic damping and vortex resonance.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/39542?expand=metadata","@language":"en"}],"FullText":[{"@value":"EXPERIMENTAL INVESTIGATION OF THE AEROELASTIC INSTABILITY OF BLUFF TWO-DIMENSIONAL CYLINDERS by PETER NOEL HAMILTON BROOKS B.A.Sc., University of B r i t i s h Columbia, 1958\" A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. i n the Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard The University of B r i t i s h Columbia August I960 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . P.N.H.Brooks. Department o f Mechanical Engineering, The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. D a t e Angnfit. 1 ? . , I960 i i ABSTRACT Aerodynamically b l u f f e l a s t i c structures such as suspension bridges and i n d u s t r i a l smokestacks have been observed to vibrate v i o l e n t l y i n the presence of a low speed wind. Although a number of theories such as Den Hartog's quasi-steady i n s t a b i l i t y c r i t e r i o n and the vortex resonance theory have been proposed to explain the phenomenon, the problem i s s t i l l not completely solved nor yet f u l l y understood. The purpose of t h i s research was to dynamically t e s t a number of two-dimensional c y l i n -ders of simple cross-section and to observe whether or not the r e s u l t s correlated with either of the two theories mentioned above. Necessary aerodynamic c o e f f i c i e n t s were obtained by the graphical integration of measured surface pressure d i s t r i b u t i o n s . Tests were performed on models mounted e l a s t i c a l l y with six degrees of freedom. General-l y only one of two modes of v i b r a t i o n was excited at a given airspeed. Dynamic response curves are presented f o r several l i g h t l y damped cylinders. For cylinders with rect-angular cross-section of length\/width (b\/h\/ greater than 0.75, v i b r a t i o n occurred at any airspeed above a certain minimum which depended on the s t r u c t u r a l damping (gallop-ing). Cylinders with b\/h less than 0.6#3 were found to vibrate over a limited range of airspeeds which always i n -cluded the c r i t i c a l v e l o c i t y f o r resonance with the period-i c formation of vortices i n the wakes. Using quasi-steady theory, i t was found that the D-section and rectangles with i i i with b\/h less than 0.633 have zero aerodynamic damping to large relative angles of attack, due to the symmetry of the wake pressures at angles of attack greater than zero. Be-cause of this the D-section is subject to galloping only when given a substantial i n i t i a l amplitude. Vortex reson-ance was observed for a l l the cylinders tested with two ex-ceptions; the reversed D-section and the D-section with the flat face ini t ia l ly at an angle of attack greater than 40\u00b0 , both of which appear to be completely stable. Measurements of the frequency of vortex formation for stationary cylin-ders gave Strouhal numbers which showed only slight varia-tion over the speed range used in the tests, However, a strong variation with b\/h was noted for the rectangles 1.0 < b\/h < 3.0. During vibration of any kind, the fre-quency of vortex formation was controlled at the frequency of vibration which in a l l cases was a natural frequency of the system being tested. An energy balance based on quasi-steady theory and neglecting structural damping yields vel-ocity-amplitude curves which give good agreement with the experimental data for the galloping D-section and the square section at various i n i t i a l angles of attack. The test re-sults indicate that the steady state aerodynamic coeffic-ients provide a useful approximation to the dynamic values; they also indicate that any theory which wil l completely predict the behaviour of such systems must include the effects of both negative aerodynamic damping and vortex resonance. CONTENTS i v Page No. INTRODUCTION 1 TEST FACILITIES AND APPARATUS 6 MEASUREMENT TECHNIQUES 10 CALIBRATION AND ACCURACY 13 REVIEW OF CURRENT THEORIES 20 TEST RESULTS 24 DISCUSSION OF RESULTS 41 CONCLUSIONS 61 APPENDICES A. Use of the Lissajous e l l i p s e f o r the determination of the frequency of an unknown signal 66 B. Wind Tunnel Corrections 68 C. The Den Hartog I n s t a b i l i t y c r i t e r i o n 70 D. D e f i n i t i o n of force coeffients 72 E. Aerodynamic c o e f f i c i e n t s f o r the D-section cylinder 73 F. Aerodynamic c o e f f i c i e n t s f o r the rectangular cylinder 77 REFERENCES 78 SUPPLEMENTARY BIBLIOGRAPHY 80 ILLUSTRATIONS 84 V ILLUSTRATIONS F i g u r e Number 1 WIND TUNNEL AERODYNAMIC OUTLINE 2 ESTIMATION OF TEST SECTION TURBULENCE LEVEL 3 TYPICAL DYNAMIC MODEL 4 MOUNTING OF DYNAMIC MODEL 5 CROSS-SECTIONS OF DYNAMIC MODELS 6 MODELS FOR WAKE WIDTH MEASUREMENTS 7 D-SECTION PRESSURE MODEL 8 LOCATION OF PRESSURE TAPS ON D-SECTION PRESSURE MODEL 9 DETAILS OF PRESSURE MODEL 10 SQUARE SECTION PRESSURE MODEL 11 LOCATION OF PRESSURE TAPS ON SQUARE SECTION MODEL 12 LOCATION OF PRESSURE TAPS ON RECTANGULAR SECTION MODEL 13 MOUNTING OF PRESSURE MODEL 14 CROSS-SECTIONS OF PRESSURE MODELS 15 MEASUREMENT OF SHEDDING FREQUENCY 16 MEASUREMENT OF AMPLITUDE BUILD-UP WITH TIME 17 CALIBRATION OF AUDIO OSCILLATOR 18 CALIBRATION OF PHOTOCELL 19 SHAPE OF PHOTOVOLTAIC CELL USED FOR AMPLITUDE MEASUREMENT 20 NON-AERODYNAMIC DAMPING FOR DYNAMIC TESTS 21 EFFECT OF STREAMWISE POSITION IN HOLE ON AMPLITUDE v i 22 ZONES OF INSTABILITY 23 PROJECTION OF C ON UPSTREAM SURFACE (<<= 0\u00b0) 24 PROJECTION OF C ON UPSTREAM SURFACE ( There have been many reports of the periodic o s c i l l a t i o n s of smokestacks as cantilevers, and \" o v a l l i n g \" of the cross-section at both s u b c r i t i c a l and s u p e r c r i t i c a l Reynolds' numbers. Two other examples which have received attention are the v i b r a t i o n of a e r i a l pipelines and submarine p e r i -scopes. I t i s generally agreed that there are at l e a s t two types of v i b r a t i o n involved i n these i n d u s t r i a l mani-fe s t a t i o n s . In the case of v i b r a t i n g transmission l i n e s , J.P.Den Hartog? d i f f e r e n t i a t e s between two forms; gallop-ing and vortex resonance. He suggests that a system subject to galloping i s unstable at any airspeed and that the vib-r a t i o n i s characterised by large amplitudes and low frequen-c i e s . The other form, vortex resonance, produces small amplitudes and high frequencies. This report w i l l deal with tests of these e f f e c t performed on two-dimensional cylinders of several cross-sections, including semi-circular, c i r c u l a r , and a wide range of rectangles. 6 TEST FACILITIES AND APPARATUS Wind Tunnel A l l the t e s t s d e s c r i b e d i n t h i s r e p o r t were per-formed i n the U n i v e r s i t y o f B r i t i s h Columbia low-speed, c l o s e d - c i r c u i t , s i n g l e r e t u r n wind t u n n e l (see F i g . 1). The t e s t s e c t i o n i s oct a g o n a l , formed by a 27 i n . x 36 i n . r e c -t a n g l e w i t h 45\u00b0 f i l l e t s , and i s 9.0 f t . i n l e n g t h . The f i l l e t s decrease from 6.0 i n . at the upstream end to 4.5 i n . at the downstream end to o f f s e t the e f f e c t of boundary l a y -er growth . The flow i s smoothed by thr e e screens p l a c e d as shown, and enters the t e s t s e c t i o n through a 7:1 c o n t r a c t i o n cone which a c c e l e r a t e s the flow and improves i t s u n i f o r m i t y . In the t e s t area, the s p a t i a l v a r i a t i o n i n v e l o c i t y i s approximately 0.25$, and the tur b u l e n c e l e v e l ^ i s l e s s than 0.5$ (see F i g . 2). The t u n n e l i s capable o f p r o v i d i n g a steady flow i n d e f i n i t e l y at an a i r s p e e d which may be v a r i e d c o n t i n u o u s l y from 4.0 f p s to 140 f p s . T e s t s were performed w i t h i n a Reynolds Number range o f 4000 to 70,000. Models Tested A l l dynamic models t e s t e d were f a b r i c a t e d from b a l s a wood w i t h a sanded f i n i s h smooth to the touch. The s p r i n g attachments were l o c a t e d as f a r apart as p o s s i b l e i n order t o o b t a i n a flow about the center-span t h a t was c l o s e l y two-dimensional. Because of t h i s c o n s i d e r a t i o n , t o g e t h e r w i t h the geometry o f the t e s t s e c t i o n , i t was found 7 that there was coincidence between the natural frequencies of the two most important modes of v i b r a t i o n . To avoid t h i s i t was necessary to add a concentrated mass to the center-span of a l l the models tested. A t y p i c a l example of a dy-namic model i s shown i n Figure 3 . Such a model weighed approximately 200 grams, 100 grams being located at mid-span. In a l l cases, the dynamic models were f r e e l y mounted on four c o i l springs (Fig\u201e. 4) allowing six degrees of freedom. In general, only two modes of v i b r a t i o n were excited; a t r a n s l a t i o n a l v i b r a t i o n perpendicular to the longitudinal axis of the model i n a plane normal to the mean flow, and a t o r s i o n a l o s c i l l a t i o n about the mid-span, also i n a plane normal to the mean flow. Other modes that were excited on p a r t i c u l a r occasions were: torsion of the cross-section; torsion about a mid-span axis i n a plane p a r a l l e l to the flow (pitching); and t r a n s l a t i o n p a r a l l e l to the longitudinal axis (diving). The same set of springs was used f o r a l l the dynamic te s t s . These springs were closely matched, weighing 44.0, 44.0, 4 3 . 7 , and 43 .7 grams respectively, and having spring constants of 1.32 l b s \/ i n . The springs were designed to have l i n e a r load-deflection c h a r a c t e r i s t i c s f o r deflections up to nine inches. The i n i t i a l extension varied between 4.0 and 4 .5 inches. A l l dynamic models were 26.63 inches i n length, allowing a clearance of 3\/16 of an inch at either end. The cross-section and dimensions of a l l dynamic models are shown i n Figure 5 . 8 A series of static models of rectangular cross-section was used to study the variation of wake properties with the ratio b\/h (see Fig. 6). These were of hardwood construction and were fitted in the test section with no end gaps. Models used for the measurement of surface pres-sures were of two types. The f i r s t , a two inch diameter D-section cylinder (see Fig. 7), was made of hardwood with a brass central section. The brass section was hollow and fitted with pressure taps machined flush with the surface. There were twenty-five pressure taps 0.020 i n . I.D. d is t r i -buted around the section (see Fig. 8 ) . On the inside of the brass section, the pressure taps were connected by 0.070 i n . O.D. copper tubing to 0.100 i n . O.D. polyethylene tubing which passed through the wooden sections to the out-side of the test section. The second type was made of aluminum box section tubing. Mid-span sections of the alum-inum were removed and replaced with brass sections similar-ly instrumented with pressure taps (see Figs. 9,10,11, & 12). Both types of pressure model were mounted in the same manner (see Fig. 13) and kept as two-dimensional as possible. As shown the models pierced the roof and floor of the test section, and the hole was sealed with card-board shields mounted flush with the inside surface. The mounting permitted the models to be adjusted to any angle of attack. Four cross-sections were tested in this manner; one D-section, and three rectangles (see Fig. 14). 9 . Experimental Apparatus For pressure measurement, two instruments were used. The f i r s t , a Betz micromanometer which could be read to + 0 . 0 0 5 mm.w.g., was used as the tunnel speed gauge. The second, a Lambrecht i n c l i n e d micromanometer, could be read to + 0 . 0 1 millimeters of alcohol on the 1 : 5 scale, and + 0 . 0 2 5 on the 1 : 2 scale. This instrument was used f o r a l l pressure measurements associated with the models. A Flow Corporation CR3-B constant resistance-r a t i o hotwire anemometer was used f o r monitoring the large scale v e l o c i t y fluctuations associated with the wakes of both stationary and vi b r a t i n g models. Used i n conjunction with the anemometer were a Heathkit A. 0 . 1 audio frequency o s c i l l a t o r modified to give frequencies down to 5 cps, and a Dumont type 3 5 0 , 1 1 0 watt cathode ray oscillograph with continuously variable sweep frequency. The o s c i l l a t o r d i a l could be read to + 0 . 2 cps at 2 5 cps and + 1 . 0 cps at 6 0 cps. An in t e r n a t i o n a l R e c t i f i e r type B-17 selenium photovoltaic c e l l was used with a Heathkit model V-7A vacuum tube voltmeter and a Brush amplifier-chart recorder combination f o r time-amplitude studies. The voltmeter could be read to + 0 . 0 0 5 v o l t s . The frequencies of vi b r a t i o n i n dynamic tests were determined by v i s u a l observation using a strobolight. The scale of t h i s instrument could be read to + 5 cpm. 10 MEASUREMENT TECHNIQUES Frequency of Vortex Formation The frequency of vortex formation i n the wake of both stationary and dynamic models was obtained by the method of Lissajous e l l i p s e s ^ (see Appendix A). A hot-wire probe was located approximately f i v e inches downstream of the model and eight inches from the centerline of the wake (see F i g . 15). At t h i s p o s i t i o n a strong signal could be picked up from the passing vo r t i c e s without the eff e c t of the f u l l spectrum of strong turbulence i n the wake. This s i g n a l was amplified by the CR3-B anemometer and put across the v e r t i c a l d e f l e c t i o n plates of an oscilloscope. The horizontal sweep voltage of the oscilloscope :was supplied by an audio frequency o s c i l l a t o r with continuously variable frequency. To obtain the predominant frequency of the signal from the hot-wire probe, the audio o s c i l l a t o r was tuned u n t i l a stable Lissajous e l l i p s e appeared on the os-ci l l o s c o p e . The frequency of the o s c i l l a t o r signal was then the same as the frequency of vortex formation on one side of the model. Vibra t i o n Amplitude In the dynamic tests performed, two types of amplitude measurement were required. F i r s t , i t was nec-essary to measure the maximum amplitude reached at a given airspeed. This was accomplished simply by v i s u a l observa-11 t i o n of the model moving across a scale on the tunnel f l o o r using a strobolight. Second, the transient amplitude build-up s t a r t i n g from rest at a given airspeed was required. To measure t h i s , a s l o t was cut i n the wall at one end of the model; a r i g i d s t r i p of aluminum was attached to t h i s end of the model and extended through the s l o t . A l i g h t cardboard shield was attached to t h i s metal s t r i p i n such a manner that i t just prevented the l i g h t from a 200w. D.C. source from f a l l i n g on a selenium photovoltaic c e l l . When the model started v i b r a t i n g , the card moved allowing l i g h t to f a l l on the c e l l . The c e l l was c a l i b r a t e d so that the v i b r a t i o n amplitude was proportional to the voltage output of the c e l l . This voltage was passed to a Brush amplifier-recorder system which provided a chart record of the ampli-tude build-up (see F i g . 16). Aerodynamic Co e f f i c i e n t s Owing to the lack of a tunnel balance i t was nec-essary to obtain l i f t and drag c o e f f i c i e n t s i n d i r e c t l y by the graphical integration of surface pressures. Although t h i s method i s tedious, l e s s accurate, and not to be rec-ommended when a balance i s available, i t did provide very useful information on the pressure d i s t r i b u t i o n on the wake side of the models which would not otherwise have been obtained. The v a r i a t i o n of t h i s pressure d i s t r i b u t i o n with angle of attack i s shown l a t e r to have great importance with regard to the s t a b i l i t y of the cylinder under observa-t i o n . The type of model used f o r these tests i s shown i n Figure 10. Two s t e e l rods were r i g i d l y attached to the 12 model outside the test section. When the model was ad-justed to a particular angle of attack, the rods were screwed directly to the panels of the test section, thus eliminating any possibility of model movement during a test. The method was simple to use and did not require any special fitt i n g s to accommodate the different cross-sections tested. In a l l cases, there were no end gaps and the system was kept as geometrically two-dimensional as possible. Wake Width The width of the wake at any point downstream of a model was defined as the cross-stream distance between two points at which the total head was ninety-five percent of the total head upstream of the model. The total head profile was taken by traversing a 0.125 i n . diameter Pitot probe across the wake. The traversing mechanism could move the probe in accurate increments down to 0.01 i n . The total head was measured on the 1:2 scale of a Lambrecht inclined manometer. Due to the large scale vortical flow in the wake, the reading from the probe was very unsteady. It was found that by inserting a choke with approximately 0.005 i n . I.D. in the lead to the manometer, a f a i r l y steady reading could be obtained. 13 CALIBRATION AND ACCURACY Tunnel Speed The Betz micromanometer used to measure the tunnel airspeed could be read to + 0.005 mm.w.g. At the lowest v e l o c i t y pressure ( p n = 0.11 mm.w.g.) there i s a possible error of 5$. This error decreases as the a i r -speed increases. The corresponding error i n v e l o c i t y i s then 2.5% f o r the worst case. The zero p o s i t i o n was check-ed a f t e r each run. This instrument i s provided with a ground glass screen on which the reading i s projected so that parallax i s prevented. A l l measurements of v e l o c i t y pressure were referred to standard temperature (520\u00b0R) and pressure (29.92 i n . hg.). Corrections due to the presence of the model i n the tunnel are l i s t e d i n Appendix B. These corrections were applied only when i t was necessary to com-pare some t e s t data with published data. In the case of a vibr a t i n g model, the corrections are not applied, since the vi b r a t i o n i t s e l f may be ch a r a c t e r i s t i c of the restrained flow. The s p a t i a l v a r i a t i o n of v e l o c i t y i n the t e s t sec-t i o n has been found to be of the order of 0.25% \u2014 a neglig-i b l e amount. I t was noted that the pressure f i e l d of the models extended upstream to the piezometer r i n g at the down-stream end of the contraction cone. When a 2 i n . D-section was mounted 24 inches downstream of the piezometer ring, the tunnel speed gauge gave a ve l o c i t y which was 1.5% low. At 14 a distance of 29 i n . the error was 1.1$ and at 34 i n . i t was 0.2$%. This correction was applied f o r a l l 2 i n . models tested. For a 1.0 i n . diameter model i t i s estimated that the upstream e f f e c t i s approximately one fourth of that f o r the 2 inch model. This was neglected. Thus the possible error i n v e l o c i t y was l e s s than 3.2$ f o r a 1.0\" model. Strouhal Number The v a r i a t i o n i n cross-section dimensions along the span was l e s s than 3$ f o r h = 1 inch, and les s than 2$ f o r 2 inch models. Curvature of the span between tangents at the extremities was l e s s than one degree. The modified Heathkit audio o s c i l l a t o r was calibr a t e d by passing the output to a Brush amplifier-recorder. The frequency of the o s c i l l a t o r signal was obtained d i r e c t l y from a chart record. This was done f o r a suitable range of frequencies and the r e s u l t s are presented i n Figure 17. The scale could be read to + 0.2 cps at 25 cps, and + 2 . 0 cps at 60 cps \u2014 the max-imum possible error i s 3.3$ at 60 cps. As stated previous-l y , the possible error i n v e l o c i t y i s les s than 3.2$. As-suming the worst condition f o r a l l variables at a v e l o c i t y pressure of 4 .0 mm.w.g., a model with h - 1 inch and S - .13 y i e l d s a maximum possible error i n S of 10$. This i s of course based on a single determination. Values of S obtained by taking a point from the average curve of shed-ding frequency against v e l o c i t y should have a maximum error comparable to that encountered i n measuring h. The error involved i n deciding when the Lissajous e l l i p s e becomes stationary i s n e g l i g i b l y small. C y c l i c movement at 0.02 cps 15 i s c l e a r l y d i s c e r n i b l e , and would cause an error of approx-imately 0.2% for a t y p i c a l measurement. The above estimate represents an upper l i m i t f o r t h i s error. Vibration Amplitude The measurement of the maximum amplitude at a given airspeed was accurate to + .05 inch. Thus f o r a 2 inch amplitude the possible error was les s than 2.5%. For time-amplitude measurement, a photocell was used. The active surface of the c e l l was masked so as to give a voltage output proportional to the length of c e l l exposed.(see F i g . 18}. This c a l i b r a t i o n was done with a Heathkit vacuum tube voltmeter which could be read to + 5 m.v. The output from the photocell was passed to a Brush amplifier-recorder systemi- This recorder was calibrated and found to give a pen d e f l e c t i o n which was d i r e c t l y pro-portional to the input voltage. Thus the amplitude of the trace on the recorder was proportional to the amplitude of vi b r a t i o n . The largest deviation from l i n e a r i t y i n the photocell c a l i b r a t i o n i s approximately 4.5%. The error i n -volved i n converting the chart amplitude to v i b r a t i o n amp-l i t u d e could be as much as 20% f o r very small amplitudes and l e s s than 10% f o r A > 1\/4: inch. Figure 19 shows the approximate shape of the active surface of the photocell. L i f t and Drag C o e f f i c i e n t s The error involved i n measuring the l i f t and drag c o e f f i c i e n t s was a combination of several possible errors. 16 (a) The error i n positioning the model i n the tunnel (b) Neglect of the viscous shear forces (c) Error i n pressure measurement (d) Error i n graphical integration (e) Error i n measuring the upstream v e l o c i t y . (a) This error i s d i f f i c u l t to i s o l a t e , and should be com-bined with any asymmetry of the basic tunnel flow. How-ever, the experimental l i f t curves a l l pass through zero at \u00b0< - 0 suggesting that there i s no asymmetry i n the flow, and that the error i n locating the model i s small. The slopes of the drag curves at \u00b0< - 0 are close to zero, so that errors i n positioning the model have l i t t l e e f f e c t . These models were located to within + 10 minutes of arc. (b) When f l u i d flows over a surface a v e l o c i t y gradient exists i n the f l u i d next to the surface. A shear stress between the f l u i d and the surface also exists which i s proportional to t h i s v e l o c i t y gradient. For a l l the dy-namic tests the separation point occurs at the upstream edge with the exception of the c i r c u l a r cylinder. Because of t h i s , the contribution of the shear to the drag i s very small (Ref. 1, p. 3-12, F i g . 22), and the neglect of t h i s contribution causes an error of l e s s than 0.5$. This i s within the accuracy of the measurement techniques. (c) Here again, the percentage error varies with the pres-sure. However, there were a large number of pressure taps, at each of which several readings were taken. This would tend to average out any random errors. (d) The error i n graphically integrating the curves of 17 pressure c o e f f i c i e n t versus model width was less than \u00b1 1%. (e) A l l pressure tests were done at 63.0 fps. The error involved f o r that speed, with h = 2 inches, i s les s than + 1.0%. Since i t i s the slope near \u00b0< = 0 rather than the ordinate of the l i f t curve that i s important we can consider the error i n taking the slope between o( = 0 \\ of = 3 degrees. For the l i f t , (b) causes a ne g l i g i b l e error at small \u2022=< and (c) i s assumed to be zero. Includ-ing the eff e c t s of (a), (d), and (e), we get errors of \u00b1 0.5%, \u00b1 1.0%, and + 1.0% respectively, a t o t a l of + 2.5%. For the drag, (a) i s small, and (c) i s again taken to be zero. For (b), (d), and (e) we get -0.5%, + 1.0%, and +1.0% respectively, or a maximum error of -2.5%. The standard wind tunnel corrections (Appendix B) are a p p l i -ed only f o r purposes of comparison with published data. Measurement of Wake Width To obtain the wake width, a t o t a l head tube was traversed across the wake and a continuous p r o f i l e was ob-tained. The d e f i n i t i o n of the wake width was arbit r a r y and the actual values of t o t a l head were unimportant. Thus any errors would be found i n the traversing mechanism and i n the model diameters. The main element of the traversing mechanism was a l \/ 2 \" x 20 screw thread. No error was found i n t h i s . For t h i s p a r t i c u l a r measurement, the model width h was accurate to within + 1%. Two-Dimensionality of Tests 18 In the MTwo-Dimensional n pressure t e s t s , the models were supported outside the tunnel so that geometric-a l l y they were two-dimensional. The pressure taps were located within one inch on either side of mid-span. In the case of the D-section and square cylinders, tests with the measuring sections located 3.0 inches from the walls gave the same r e s u l t s as the mid-span t e s t s . This, together with the f a c t that the force c o e f f i c i e n t s obtained from integrating the surface pressures agreed cl o s e l y with pub-li s h e d r e s u l t s , indicated that the flow also was two-dimensional. In the dynamic tests i t was very d i f f i c u l t to obtain two-dimensional conditions. I t was decided that mounting the model with springs inside the tunnel and a 3\/16 inch gap at either end had l e s s e f f e c t on the two-dimensionality of the flow than mounting the springs on the outside and allowing the model to pass through holes large enough to permit the v i b r a t i o n amplitudes encountered. Dy-namic tests with end gaps of l\/4 i n . , 1\/2 i n . , and 3\/4 i n . showed no appreciable change i n the amplitude at a given v e l o c i t y , whereas allowing the model to extend through a 6.0 i n . diameter hole produced a completely d i f f e r e n t set of v i b r a t i o n c h a r a c t e r i s t i c s . Structural Damping In a l l the dynamic tests the s t r u c t u r a l damping 19 was kept constant i n that the same springs and mounts were used throughout. The damping was measured by obtaining the time-amplitude decay of a f l a t plate of comparable mass i n s t i l l a i r ( F i g . 2 0 ) . I t should be noted that at large amp-li t u d e s the spring forces became large enough to cause the tunnel walls to vibrate appreciably. This e f f e c t would make the actual damping greater than the measured value f o r the higher airspeeds and would tend to reduce the amplitude reached at a given airspeed. It i s also noted that as the airspeed increases the v i b r a t i o n caused by the tunnel fan increases. Since the dynamic tests were performed at r e l -a t i v e l y low airspeeds, t h i s e f f e c t should be small. 20 REVIEW OF CURRENT THEORIES Den Hartog's \"Negative Slope\" 1 Theory? This theory considers an e l a s t i c a l l y mounted c y l i n d r i c a l body with negative aerodynamic damping; i . e . the cross-sectional shape of the cylinder i s such that, i f the body i s moving downward i n the presence of a steady wind, i t w i l l experience a downward force due to the wind. Den Hartog expresses t h i s condition f o r i n s t a b i l i t y i n a simple r e l a t i o n s h i p between the steady state l i f t and drag c o e f f i c i e n t s f o r the cross-section (see Appendix C). He shows that i f (dC^\/d* ) ^ + C^0 <0, then the cross-section w i l l be unstable. Thus a cross-section which s a t i s f i e s t h i s c r i t e r i o n should be unstable at a l l airspeeds. The lowest airspeed at which v i b r a t i o n w i l l occur should depend only on the s t r u c t u r a l damping, and the amplitude of v i b r a t i o n should depend on the a i r v e l o c i t y ; i . e . , a maximum ampli-tude f o r a given a i r v e l o c i t y should be reached when the net input of energy per cycle from the wind i s just equal to the energy dissipated by s t r u c t u r a l damping. Cross-sections which do not s a t i s f y t h i s c r i t e r i o n should be aero-dynamically stable. It should be noted that i t has yet to be proved that the steady state aerodynamic c o e f f i c i e n t s can be used to replace the instantaneous dynamic c o e f f i c i e n t s . P a r k i n s o n ^ has obtained a solution for the non-l i n e a r d i f f e r e n t i a l equation of motion that r e s u l t s from 21 Den Hartog's c r i t e r i o n when the steady state aerodynamic c o e f f i c i e n t s are expressed as polynomial functions. His r e s u l t s f o r a square section cylinder are presented l a t e r on. 12 D.B. Steinman has put forward a theory which also makes use of the steady state aerodynamic c o e f f i c -i e n t s . He also disregards any effect of vortex formation other than the fac t that the aerodynamic c o e f f i c i e n t s are influenced by the time-average of the vortex induced pres-sures. This theory was developed to predict the v i b r a t i o n c h a r a c t e r i s t i c s of a wide section, such as a bridge deck, but can be applied to any aeroelastic system. The Vortex Theory When f l u i d flows past a b l u f f c y l i n d r i c a l object separation occurs and two shear layers are formed. These surfaces of discontinuity are extremely unstable and break up to form large vortices centered approximately on the shear layers. I t was observed by S t r o u h a l ^ that the f r e -quency f at which the vor t i c e s are formed on each side i s d i r e c t l y proportional to the f l u i d v e l o c i t y Vg, and inversely proportional to the cylinder diameter h . This r e l a t i o n was expressed by Strouhal i n the form of a dimen-sionless parameter, the Strouhal Number S - fh\/Voo (1) Upstream of the cylinder the flow i s assumed to be i r -22 r o t a t i o n a l , so that when a vortex i s formed i n the wake, then, i n accordance with Helmholtz 1 Laws12*\", c i r c u l a t i o n of opposite d i r e c t i o n i s set up about the cylinder. The in t e r a c t i o n of t h i s c i r c u l a t i o n with the mean flow causes a force to act on the body i n a d i r e c t i o n normal to the flow (Kutta-Joukowski 1 2 f). This vortex induced force w i l l alternate at the frequency of vortex formation. The vor-tex theory at t r i b u t e s the aerodynamic ex c i t a t i o n of an e l a s t i c a l l y mounted cylinder to the action of t h i s period-i c force having a certain degree of resonance with one of the natural modes of vi b r a t i o n of the system. I f the low-est natural frequency of the system i s f p , then the sys-tem should s t a r t v ibrating near a c r i t i c a l v e l o c i t y V Cp where V c p = f ph\/S (2) Once the vib r a t i o n has started i t controls the frequency of formation of the v o r t i c e s so that two vortices are formed every cycle. In t h i s way the v i b r a t i o n remains i n a condition of resonance over a range of airspeeds. As the airspeed i s increased, the vor t i c e s grow stronger and the amplitude increases, u n t i l f i n a l l y the vi b r a t i o n loses control of the vortex action and dies down. At t h i s point the frequency i s again determined by the v e l o c i t y , the Strouhal number, and the diameter. V i b r a t i o n w i l l reappear at the next c r i t i c a l v e l o c i t y . This theory indicates that e l a s t i c a l l y mounted 23 cylinders of any cross-section should be excited within cer t a i n ranges of airspeed i f the damping i s not too high. Here the amplitude of v i b r a t i o n w i l l depend on both the s t r u c t u r a l and aerodynamic damping, the shape of the cross-section, and the a i r v e l o c i t y . There should also be ranges of airspeed i n which such systems are not excited. F l u t t e r The f l u t t e r type of aerodynamic e x c i t a t i o n d i f -f e r s from those mentioned previously i n that two degrees of freedom are excited simultaneously instead of just one. Usually t h i s takes the form of a transverse v i b r a t i o n coupled with twisting of the cross-section. Generally, because the exciting forces are not small compared to the e l a s t i c and i n e r t i a forces, the two natural frequencies w i l l d i f f e r from those measured i n s t i l l a i r . Since the aerodynamic forces vary -with the wind speed, so w i l l the natural frequencies and a c r i t i c a l speed can be reached at which they coincide. When t h i s occurs the r e s u l t w i l l be a very rapid buildup to large amplitudes. 24 TEST RESULTS D-Section \u2014 Open Holes at Ends Preliminary dynamic tes t i n g was performed on a 2\" D-section cylinder p i e r c i n g the f l o o r and roof through 6.5 inch c i r c u l a r holes and suspended on four c o i l springs placed outside the tunnel. It was noted f o r a given a i r -speed that the r e s u l t i n g amplitude of v i b r a t i o n i n the plunging mode was a function of the streamwise p o s i t i o n of the model i n the hole. Because of the awkward geometry of the c i r c u l a r hole, i t was decided to investigate t h i s var-i a t i o n further by using rectangular openings. The model was tested with 8\" x 10\" openings and the same effect was observed. Some degree of c o r r e l a t i o n was obtained between the two cases by p l o t t i n g the amplitude of v i b r a t i o n ver-sus the r a t i o A\/A Q, where A i s the area of the opening upstream of the model and'. & Q i s the area upstream of the model when the reduced amplitude A p i s 0.025. The re-s u l t s are shown i n Figure 21. Both sets of data plot on a single curve. The aeroelastic behaviour over a wide range of airspeeds i s shown i n Figure 22. Three modes of v i b r a -t i o n were observed: plunging, a t r a n s l a t i o n a l v i b r a t i o n perpendicular to the longitudinal axis of the model and i n a plane normal to the mean flow; diving, a t r a n s l a t i o n a l v i b r a t i o n p a r a l l e l to the longitudinal axis; and pitching, a t o r s i o n a l v i b r a t i o n about a mid-span axis i n a plane 2 5 p a r a l l e l to the flow. Below 6 2 . 0 fps, and for any posi-t i o n i n the hole, the plunging mode occurs. At higher airspeeds, both pitching and diving modes occur. This wide range of airspeeds over which plunging occurred suggests that the D-section shows i n s t a b i l i t y of the \"galloping\" type. However, at an airspeed of 6 6 . 0 fps, plunging occurred at X = 9 . 8 7 5 inches but not at X \u00ab= 8 . 5 0 inches. This suggests that Den Hartog's i n s t a b i l i t y c r i t e r i o n (Appendix C) f o r t h i s section should be a function of the hole p o s i t i o n at t h i s p a r t i c u l a r airspeed, i . e . : at X = 9 . 8 7 5 inches (dCFv\/d=< ) ^ _ Q < 0 ( 3 ) and at X = 8 . 5 inches UCFv\/d<< ) < a = Q > 0 ( 4 ) In order to check t h i s , a s t a t i c model was con-structed and instrumented to measure surface pressures. The pressure c o e f f i c i e n t C p was measured at the mid-span f o r both positions i n the hole (see Figs. 2 3 , 2 4 ) . Both showed the same C p v a r i a t i o n on the upstream face and constant wake pressures with respect to the chord, the value being s l i g h t l y lower f o r X = 9 . 8 7 5 inches. The sec-t i o n a l l i f t and drag c o e f f i c i e n t s (see F i g . 2 5 ) were ob-tained by graphical integration and the aerodynamic damping force c o e f f i c i e n t C F y was calculated. I t was found f o r both cases that (dGFv\/d\u00b0< ) ^ _ 0 = 0 within the accuracy of the measurement techniques. Thus Den Hartog's c r i t e r i o n 26 cannot be applied here. I t i s shown (Appendix E) that the damping force c o e f f i c i e n t Cp v i s exactly zero i f the wake pressure i s constant or symmetrical with respect to the chord f o r \u00b0<> 0\u00b0 . The l i f t and drag c o e f f i c i e n t s ob-tained were approximately 30% lower than values quoted by Cheers^. This was due to the outflow through the end holes from points upstream of the model, and the inflow to the wake downstream of the model. Because of the obvious e f f e c t of the holes on the e l a s t i c behaviour of the system, i t was decided to eliminate the holes, to place the springs i n the airstream (as close to the ends as possible) and to allow only a small gap at the ends of the model. I t was f e l t that the eff e c t of the springs on the flow would be small compared to that of the holes. Two-Dimensional D-Section Pressure tests were performed on a 2\" D-section at various angles of attack with the end holes completely sealed. A check, with the measuring section three inches from the f l o o r , showed that the flow was two-dimensional. Typical C p d i s t r i b u t i o n s are shown i n Figures 26, 27. The upstream v a r i a t i o n at *C - 0\u00b0 shows good agreement with that of a f l a t plate given by Fage^. The wake pressure at \u00b0\u00a3 = 0\u00b0 i s no longer constant but i s s t i l l symmetrical with respect to the chord. This symmetry i n the wake per-s i s t s up to \u00b0C = 25\u00b0 while the stagnation point upstream moves toward the leading edge. At <= 35\u00b0 the wake pres-sure has become noticeably asymmetric, and at 40\u00b0 there 27 i s a sudden r i s e i n the l e v e l of the wake pressure and a suction peak forms downstream of the leading edge, i n d i c -ating that the flow has re-attached. The sectional l i f t and drag c o e f f i c i e n t s were ob-tained again by graphical integration and are shown i n Figures 28 and 2 9 . The l i f t curve, uncorrected f o r wind tunnel e f f e c t s , shows values of l i f t approximately 13$ higher than those presented by Cheers1-* f o r G < < 3 5 \u00b0 . However, Cheers data reaches a peak at \u00ab=< = 40\u00b0 which was not found i n the present t e s t . The data corrected f o r wind tunnel e f f e c t s show better agreement with Cheers' data over most of the range. It should be noted that small end holes we,re permitted i n the tests done by Cheers which would have the e f f e c t of reducing the slope of the l i f t curve. The other curve presented i s that of -Cdtan\u00b0< . I t i s shown i n Appendix E that i f the wake pressure i s symmetrical with respect to the chord f o r <<= 0 \u00b0 , then - -C dtan\u00ab (5) This curve follows the experimental l i f t data very closely up to 2 5 \u00b0 , the angle of attack at-which the wake pressure becomes asymmetric. Thus fo r the two-dimensional D-section (dCj>v\/d<* )o<\u00b0<<250 = 0 , i . e . , Den Hartog's c r i t e r i o n i s not s a t i s f i e d , and the D-section should not gallop when re-leased from r e s t . Cheers 1^ finds that the D-section i s i n i t i a l l y stable, and H a r r i s 1 ^ finds that i t i s unstable. The drag ( F i g . 29) i s also found to be higher than that 28 given by Cheers. A s t a t i c pressure traverse along the center l i n e of the wake shows a gradual increase i n suction to a maxi-mum at approximately 1.25 diameters downstream of the f l a t face (see F i g . 30). These data were obtained by placing a s t a t i c probe i n the wake p a r a l l e l to the flow. Because of the v o r t i c a l cross flow, the pressure l e v e l read i s lower than the actual value. The readings are adjusted by a constant f a c t o r to the pressure at the cylinder surface obtained from previous t e s t s and show a d i s t r i b u t i o n which i s s i m i l a r to that obtained by Roshko\"^ f o r the c i r c u l a r cylinder. However, Roshko's r e s u l t s f o r a f l a t plate show the point of vortex formation to be approximately 2.0 dia-meters downstream. I t i s noted that the pressure at the surface of the plate i s higher than that obtained f o r the D-section. This suggests that Roshko's tests may have been performed with small end gaps which would have i n -creased the pressure and which may also have affected the point of vortex formation. To check t h i s , the wake t r a -verse was repeated f o r the D-section cylinder with the 8\" x 10\" holes open. I t was found that there was only a s l i g h t decrease i n pressure to a minimum at approximately 2.75 diameters downstream of the f l a t face, and that the general pressure l e v e l was much higher than f o r the two-dimensional case. I f the holes at the ends of the model are open, the inflow to the wake tends to suppress the pressure gradient along the wake i n that the maximum inflow i s directed toward the point of minimum pressure. I t i s suggested that the suction peak i n the wake close to the afterbody of the D-section causes the surface pressure p r o f i l e to achieve a minimum f o r the two-dimensional case, whereas the surface pressure for the 'open-end' case i s not affected i n t h i s way and the p r o f i l e i s f l a t (see Figs. 23, 26). In the dynamic tests on the two-dimensional D-section only two modes of v i b r a t i o n were excited; (a) a t r a n s l a t i o n a l o s c i l l a t i o n perpendicular to the longitud-i n a l axis of the model and i n a plane normal to the flow, hereafter referred to as 'plunging', and (b) a t o r s i o n a l o s c i l l a t i o n about the mid-span also i n a plane normal to the flow and hereafter referred to as ' t o r s i o n 1 . The model tested had the following c h a r a c t e r i s t i c s : Diameter - 2.0 i n s . Natural frequency i n plunging - 8.5 cps Natural frequency i n t o r s i o n - 10.35 cps Weight - 233 gms. The dynamic response f o r t h i s D-section at 3 = 0 \u00b0 , where B i s the i n i t i a l angle of attack of the section, i s shown i n Figures 31 and 32. Two sets of data are presented -the r a t i o of amplitude to diameter and the r a t i o of the frequency of formation of vortices downstream of one edge of the model to the natural frequencies i n plunging and t o r s i o n . The abscissa i s the r a t i o of the upstream velo-c i t y to the \" c r i t i c a l \" v e l o c i t y , where V c p and V c t 30 are defined as follows: V c p - f p h \/ S { 6 ) V c t = f th\/S (7) where h i s the diameter, S the Strouhal number, and fp and the two natural frequencies. The dynamic response may be described as follows: a) I f the model i s restrained at an airspeed at which i t i s known to plunge, the frequency of vortex formation w i l l be determined by the Strouhal number. b) When i t i s released, plunging o s c i l l a t i o n begins and gradually builds up. I t requires no i n i t i a l amplitude. c) At an amplitude estimated at A p = 0.15, the vortex frequency changes abruptly to the plunging natural f r e -quency. d) The rate of build-up increases and a maximum amplitude i s reached. e) I f the v i b r a t i o n i s stopped, the frequency reverts to the Strouhal frequency. The control of the vortex frequency w i l l be referred to as \"capture\". For the model t e s t s , v i b r a t i o n began at V n c\/V Cp - 0.54 with capture occurring at one h a l f the natural frequency. This continued to V^\/V^p = 0.71 at which speed the captured vortex frequency jumped between 0.5 f p and f p . From V n c \/ V c p = 0.71 to 1.07, A p increas-ed to a maximum, and capture was maintained at fp. Be-3 1 yond 1 . 0 7 a mixture of plunging and torsion occurred. Both the t o r s i o n a l and plunging reduced amplitudes are defined as the r a t i o of the t i p amplitude A to the model width h. In order to provide a basis f o r comparing the two amplitudes, one can consider the poten t i a l energy stored at maximum amplitude f o r both cases. I f t h i s i s done we f i n d that the t o r s i o n a l amplitude involving the same pot e n t i a l energy as the plunging amplitude Ap i s given by where \/ i s the length of the model and jts i s the distance between spring mounts. For a l l dynamic tests performed | s = 0 . 5 2 5 . At v\"nc\/Vc-t = 0 . 9 5 pure torsion occurred accompanied by capture at f t . A maximum amplitude was reached at V n c\/V c-t <= 1 . 0 5 above which the amplitude de-cayed rapidly u n t i l at V n c \/ V c t = 1 . 2 there was no vibra-t i o n ; Below V n c\/V Cp = 0 . 5 4 , and above V n c \/ V c t - 1 . 2 , the only motion observed was a random buffeting and i f the model was released from an i n i t i a l displacement the vibra-t i o n would steadily die out. Dynamic te s t s were next performed on a two inch D-section f o r a wide range of 3 (see F i g . 3 3 ) . The vib r a -t i o n c h a r a c t e r i s t i c s obtained were si m i l a r to those f o r the model i n i t i a l l y at P = 0 , except that the maximum amp-li t u d e s decreased steadily with increasing B u n t i l at P = 4 0 \u00b0 no vi b r a t i o n occurred. The i n s t a b i l i t y c r i t e r i o n , 32 dCp \u00ab= 3 . 5 . From b\/h = 2.9 to 4.45, ^ remains approximately constant. There i s no abrupt change i n & corresponding to the r i s e i n S be-tween b\/h = 2.4 and 3. 40 Measurements of the time-amplitude response were made on the square section cylinder. Figure 49 shows the response curve at V n e =4.74 V Cp and indicates that the model w i l l achieve i t s maximum amplitude approximately 250 cycles a f t e r being released from r e s t . The shape of t h i s curve i s s i m i l a r for a l l airspeeds but the time to maximum amplitude varies with V n c\/V Gp. Similar measure>-ments were made on a square section cylinder f o r which the st r u c t u r a l damping was increased approximately by a factor of four. The ef f e c t of t h i s was to s h i f t the v e l o c i t y at which the vi b r a t i o n i n i t i a t e s to a higher value (Fig. 39). The time required to reach maximum amplitude at various airspeeds was measured f o r the two l e v e l s of st r u c t u r a l damping (Fig. 50). The curve f o r the more highly damped case i s displaced to the rig h t of the other and tends to i n f i n i t y f o r V n c =3.8 V Cp. The other curve shows the same trend but at V n c = 2 V Cp the number of cycles re-quired to b u i l d up to maximum amplitude begins to decrease and the lowest value reached i s approximately 80 cycles at V-nr. = 1.2 V r r i. 41 DISCUSSION OF RESULTS D-Section Cylinder It i s noted i n the case of the D-section passing through holes i n the roof and f l o o r that a form of gallop-ing occurs. (dCFV\/d\u00b0< ) ^ _ 0 was measured f o r the midspan and found to be exactly zero. However, there were strong end flows and (dGpv\/d<=< f o r the whole model may have been negative. It was not possible to measure t h i s because the pressure taps were not a l l i n the same horizontal plane and the spanwise pressure gradient was large. In a s i m i l a r way the desk model D-section proposed by Den Hartog may s a t i s f y the galloping c r i t e r i o n . The f a c t that dF\/d^ be-comes negative at << = 25\u00b0 agrees with the observed f a c t 2 that Lanchester's t o u r b i l l i o n does auto-rotate i f given a substantial i n i t i a l spin. During v i b r a t i o n the D-section appears to move i n a plane normal to the flow. I f the cross-section twists about the longitudinal axis i t does so with an amplitude that i s not d i s c e r n i b l e ; i . e . , l e s s than one h a l f of one degree. I t i s f e l t that the observed vibrations are \"pure M : plunging or torsion, and not a f l u t t e r type combination. This i s borne out by the f a c t that the systems always vib-rate at t h e i r natural frequencies over the entire range of airspeed. As the pressure tests suggest, the response i s 42 not of the \"galloping\" type but rather a form of extended resonance with the vortex formation. Capture at 0.5 fp corresponds to resonance with the f i r s t harmonic, and at fp and f ^ to fundamental resonance. Resonance with the f i r s t harmonic i s d i f f i c u l t to reconcile with the physical picture since i t implies that a vortex i s formed every second cycle. I t i s thought that during capture at fp, one vortex i s formed just a f t e r the model reaches the max-imum amplitude, thus inducing an equal and opposite c i r c u l -a tion about the model and causing a force to act on i t i n the d i r e c t i o n of the motion. For the D-section, the aerodynamic damping i s zero up to <^ = 25\u00b0. For 25\u00b0 < \u00b0< <( 40\u00b0 i t i s negative and for 1.02. This agrees with the res u l t obtained i n the previous t e s t , as does the speed at which plunging f i r s t s t a r t s , i . e . close to 0.5 V Cp. I t i s noted that the amplitudes are considerably larger f o r a given value of V n c\/V Cp (F i g . 31). In the present t e s t the spring constant of the e l a s t i c system and the mass of the model were less than i n the previous t e s t , however fp was approximately the same. With the l i g h t e r system, le s s energy was trans-ferred to the tunnel than i n the previous test i n which the v i b r a t i o n of the tunnel walls, caused by the greater spring forces, became appreciable at the larger amplitudes. It i s f e l t that the difference i n amplitudes observed i n the two tests i s att r i b u t a b l e to t h i s . C i r c u l a r Cylinder The c i r c u l a r cylinder has p o s i t i v e aerodynamic damping and i s subject only to fundamental resonance with the formation of vor t i c e s i n i t s wake. The dynamic re-sponse i s t y p i c a l of a resonance phenomenon, and d i f f e r s from the D-section i n that i t i s excited over a smaller range of airspeeds and the amplitude curve shows a more l o c a l i z e d peak. Reversed D-Section This section showed no i n s t a b i l i t y at any a i r -speed including V n c = V Cp and V c^. I t i s expected that a l l e l a s t i c structures which shed vortices i n a steady airstream should show some exci t a t i o n when the f r e -quency of vortex formation approaches a natural frequency of the structure. However, there are two reasons why the reversed D-section would tend to be stable. F i r s t , the aerodynamic damping i s high and p o s i t i v e . Like the c i r -cular cylinder, C L i s zero f o r \u00b0<. \u00ab= + 20\u00b0, thus G F v \" CDtan\u00b0< Sec\u00b0< (14) and f o r small \u00b0< dC F v\/d\u00b0< = C D - 1.15 (Cheers 1 5) Second, the reversed D-section, unlike any other section tested, has no afterbody extending into the wake. Since the point of vortex formation i s determined by the point 46 of flow separation, the reversed D-section w i l l be re-l a t i v e l y further from the point of vortex formation and from the entire vortex street than any section with an afterbody. This e f f e c t may be s u f f i c i e n t to reduce the exci t i n g force to a l e v e l that i s i n s u f f i c i e n t to over-come the s t r u c t u r a l and aerodynamic damping. The Rectangular Cylinder The rectangular cylinder shows both galloping and vortex-induced v i b r a t i o n . For small values of b\/h the wake pressure remains symmetrical with respect to the chord f o r \u00b0< > 0, and consequently the aerodynamic damping i s exactly zero f o r a wide range of \u00b0< . For larger values of b\/h the wake pressure becomes asymmetric at small \u00b0< and the damping becomes negative. However, as b\/h increases, the angle at which the aerodynamic damping becomes p o s i t i v e decreases and hence the plunging amplitude at a given airspeed decreases. A s i m i l a r ef-fe c t was noticed i n te s t s on a square section cylinder at various i n i t i a l angles of attack 6. As 3 i s i n -creased, the range of \u00b0< over which negative damping occurs decreases, and the possible amplitude at a given airspeed also decreases. A l l the amplitude curves (Fig.37) appear to st a r t close to the c r i t i c a l v e l o c i t y . This i s a c h a r a c t e r i s t i c of the galloping type of v i b r a t i o n ! 9 . The airspeed at which v i b r a t i o n w i l l s t a r t usually de-pends on the s t r u c t u r a l damping, but below a certa i n l e v e l the s t a r t i n g speed becomes independent of the damping and 47 remains fixed at the c r i t i c a l v e l o c i t y V Cp. For the part-i c u l a r l e v e l of damping the vi b r a t i o n i s started by vor-tex resonance and at a s l i g h t l y higher airspeed i t combines with galloping. As the airspeed increases the i n i t i a l vortex e f f e c t decreases, and the v i b r a t i o n tends toward pure galloping. For an e l a s t i c system with greater struc-t u r a l damping (Figs. 38, 39) the curves f o r b\/h < 0.68 would be grouped about V n c = V Cp as before. For b\/h >0.75 vortex induced plunging would also occur at V n c = V Cp, but t h i s would not change to galloping. It would die out as V n c increases u n t i l f i n a l l y at a s u f f i c i e n t l y high V n c the galloping type of vib r a t i o n would s t a r t . For b\/h> 2.5 the rectangle starts to behave l i k e a f l a t plate. The natural frequency of torsion of the cross-section decreases and exposes that mode of vi b r a t i o n to vortex e x c i t a t i o n within the range of airspeeds used. It should be noted that these t o r s i o n a l modes were pure and there was no suggestion of a f l u t t e r type combination with plunging, and no v a r i a t i o n of frequency with airspeed. When the. rectangular cylinder was galloping, capture occurred at speeds up to V n c = 7 V Cp, the highest used i n these t e s t s . I t w i l l therefore be necessary to i n -clude the eff e c t s of vortex formation i n any theory that w i l l describe the motion completely. Roshko 1? has proposed a wake Strouhal number S* defined by 48 S* = nd\u00ab\/U s where n i s the vortex frequency, d* i s the distance between the two shear layers, and U s i s v e l o c i t y at the edge of the wake. He proposes that S* should have a constant value f o r a l l b l u f f bodies. In his paper he refers to tests done on a c i r c u l a r cylinder with a s p l i t t e r plate i n the wake. As the s p l i t t e r plate i s moved downstream, i t forces the vortices to form further from the cylinder base. The r e s u l t of t h i s i s to decrease the vortex f r e -quency n, and the v e l o c i t y U s, and increase the wake width d*, and to maintain a roughly constant S . As the distance between the s p l i t t e r plate and the cylinder i s increased, a point i s reached at which the vortices suddenly s t a r t forming upstream of the s p l i t t e r plate. When t h i s occurs the e f f e c t of the plate on the flow i s substantially reduced. The frequency, v e l o c i t y , and wake width abruptly return to values which are close to the o r i g i n a l ones and i s maintained. A similar phenomenon may occur with the rectangular cylinder. In terms of the te s t variables, S* i s defined as S* = f a i h\/kV n c (16) where k = ^ 1 - C p w and C p w i s the pressure c o e f f i c i e n t on the downstream surface of the cylinder. Now as b\/h i n -creases, f a decreases, and G p w becomes les s negative, causing k to decrease. However, unlike Roshko's case, o also decreases, and so does S*. . Values of S* f o r 49 b\/h = 0 . 5 , 1.0, and 2.0 are 0 . 4 7 , 0 . 4 6 , and 0.28 respect-i v e l y . Between b\/h = 2 . 4 and 3.0, f a suddenly increases. I t i s suggested that t h i s change i s caused by the s h i f t i n the point of vortex formation from downstream of the t r a i l -ing edge to a point between the leading and t r a i l i n g edges, s i m i l a r to the s h i f t encountered with the c i r c u l a r cylinder and s p l i t t e r plate. I t should be mentioned while r e f e r r i n g to Roshko's work that the mathematical models proposed by 20 himself and Kirchhoff would show no dynamic i n s t a b i l i t y other than vortex e x c i t a t i o n i n that the wake pressure i s considered to be constant, i . e . : dCpy\/d* = 0. Observations of the time required f o r the vibra-t i o n to build up at a given airspeed show c l e a r l y the ef-fe c t of vortex ex c i t a t i o n and s t r u c t u r a l damping on gallop-ing (Fig. 50). The test with increased s t r u c t u r a l damp-ing i s not influenced by vortex ex c i t a t i o n other than the capture which occurs a f t e r galloping has started. As V n c decreases toward the speed at which galloping s t a r t s , the time to maximum amplitude approaches i n f i n i t y . However, fo r the l i g h t l y damped case, the speed at which galloping s t a r t s coincides with the region of maximum vortex e x c i t -ation, Thus as V n c decreases, the time to build-up i n -creases u n t i l i t comes under the influence of the vortex exci t a t i o n , a f t e r which i t decreases rapidly as V n c approaches V . 5G The slope of the time amplitude curve (Fig. 4 9 ) i s of in t e r e s t because of the abrupt manner i n which i t decreases near the maximum amplitude. I t i s suggested that t h i s corresponds to the sudden onset of high p o s i t i v e damp-ing when a certain value of \u00b0C i s reached. This value of c< can be determined by obtaining an energy balance f o r a l i m i t cycle of the vibr a t i o n ; i . e . , the net energy trans-ferred from the airstream to the e l a s t i c structure should be exactly equal to the energy dissipated by the str u c t u r a l damping. An approximate energy balance i s presented. Approximate Theory f o r Galloping For a given , the aerodynamic damping force, F v , i s d i r e c t l y proportional to the square of the airspeed V and the st r u c t u r a l damping i s taken to be constant over the range of airspeeds considered. I f the system i s l i g h t -l y damped, then at high airspeeds the aerodynamic damping forces w i l l be much greater than the s t r u c t u r a l damping forces and the l a t t e r can be neglected. In the analysis given here, the s t r u c t u r a l damping i s assumed to be neg-l i g i b l y small. This assumption obviously w i l l not be v a l i d at low airspeeds, and agreement i s not expected. During galloping, i t i s found that the phenomenon of capture occurs. This periodic vortex formation w i l l have some ef f e c t on the maximum amplitude reached at a given airspeed. The strength ( P ) ' o f the i n d i v i d u a l v ortices formed w i l l vary d i r e c t l y with the airspeed, 51 and inversely with the frequency of formation. During capture, the frequency of formation i s fixed at the natur-a l frequency of the system, and therefore P w i l l depend only on the v e l o c i t y VQQ . The Kutta Jowkowski Law1**' gives the force on a body with c i r c u l a t i o n T i n a uniform airstream of v e l o c i t y V\u2014, , P = a r (17) Assuming that Helmholtz* Laws apply i n t h i s case, the force on a model due to vortex exc i t a t i o n w i l l be P\u00b0C ? V 2 (Id) Thus both the vortex exc i t a t i o n and the negative damping o e x c i t a t i o n vary with V and should therefore remain i n a constant r a t i o to one another. Since we are neglecting both st r u c t u r a l damping and vortex excitation, the v i b r a t i o n w i l l b u i l d up from rest u n t i l the r e l a t i v e angle of attack reaches the angle 4 defined by equation 13. Since the s t r u c t u r a l damping i s taken to be zero, \u00a3 i s independent of V and there-fore the amplitude w i l l be d i r e c t l y proportional to V. Thus the analysis presented here should give a v e l o c i t y -amplitude curve as shown i n Figure 51. I t i s to be expected i n the experimental case that the s t r u c t u r a l damping w i l l prevent v i b r a t i o n u n t i l a ce r t a i n minimum v e l o c i t y i s reached at which v e l o c i t y the aerodynamic damping forces become comparable to the st r u c t u r a l damping 52 forces. The broken l i n e represents the amplitude re-sponse i f the vortex excitation due to capture i s includ-ed, and i s also the asymptote to which the experimental curve w i l l tend. The position of t h i s l i n e r e l a t i v e to the l i n e representing aerodynamic damping only w i l l de-pend on the natural frequency i n two ways. F i r s t , a sys-tem with large f p should show less e f f e c t of vortex ex-c i t a t i o n than one f o r which f p i s small, and second, the quasi-steady theory i s l e s s applicable i f f i s large. Consider a spring-mass system placed i n a uni-form horizontal airstream as shown (Fig. 52). The mass M i s assumed to be p e r f e c t l y e l a s t i c . By d e f i n i t i o n CFv(CF v(o<)dy \u00ab= 0 (20) Let cot = 6 Then (* C F v ( <* )ACos6de = 0 (21) Now o( = Arctan y\/V = Arctan(A i n radians\/second It can be seen from equation (26) that the amplitude i s completely independent of the size of the v i b r a t i n g mass. 55 Velocity-amplitude curves were obtained ex-perimentally f o r the following square sections at (3 = 0 \u00b0 . 1.00\" x 1.00\" u> \u00ab 42.4 rad\/sec. 1.00\" x 1.00\" CA) = 56.0 rad\/sec. 0.75\" x 0.75\" <-> = 52.4 rad\/sec. 0.75\" x 0.75\" 70.1 rad\/sec. The data are compared with the theory and are found to give very good agreement at the higher v e l o c i t i e s (Figs.53 ,54 ) . This may be interpreted i n two ways. F i r s t the vortex e x c i t a t i o n may be very small compared to that of the neg-ative damping. Second, the s t r u c t u r a l damping may s t i l l be having an e f f e c t on the amplitude, i . e . , at higher v e l o c i t i e s the experimental curve may cross the theoret-i c a l curve. I t i s i n t e r e s t i n g to note that the system with the highest natural frequency shows the largest de-v i a t i o n from the theory. The lack of dependence of the amplitude on the size of the model i s emphasized by p l o t t i n g A versus V\/oo f o r a l l the models tested (Fig . 5 5 ) . I f equation (22) i s integrated between + \u00a3 , the second term i n equation (25) drops out, and we get; (dC F v\/cK)(Yl - (CoUTan*) 2 }d\u00ab = 0 (27) Applying equation (27) to the square i n i t i a l l y at 5\u00b0 and 9 \u00b0 , i t i s found to be s a t i s f i e d by 6 = 15.2\u00b0and e= 9.0\u00b0 56 respectively. These values were substituted i n equation (26) and the curves are compared with experimental data (Fi g . 56). For the airspeeds covered, the t h e o r e t i c a l curve gives larger amplitudes f o r both values of (3. This i s possibly due to the fac t that as (3 increases the energy involved i n negative aerodynamic damping at a given a i r -speed decreases and the effects of the st r u c t u r a l damping w i l l be noticeable to higher v e l o c i t i e s . I f (25) i s applied to the D-section, it\u00bbis s a t i s f i e d by 6 = 56\u00b0. Thus f o r V c p = 10.8 fps, ^ p = 55 rad\/sec, and h = 2 inches, equation (13) gives A p = 1.75 Vnc\/^cp* This curve i s compared with the experimental response f o r the gallop-ing D-section (Fig. 57) and shows very good agreement. General Discussion For a l l the models tested, regardless of cross-section, the observed vibrations were r e s t r i c t e d to a single plane normal to the flow, except at high v e l o c i t i e s when the streamwise d e f l e c t i o n of the model due to the drag force became appreciable. In a l l cases, the wind induced vibra-tions occurred at frequencies which corresponded to the natural frequencies obtained i n s t i l l a i r . Mixed torsion and plunging did occur but only when the frequency of the vortex formation was halfway between the two natural f r e -quencies. I t i s f e l t that f l u t t e r i s not the mechanism governing the observed vibrations. The test r e s u l t s supported to a certain extent 57 the vortex theory for the ex c i t a t i o n of b l u f f cylinders. The D-section, the c i r c u l a r cylinder, and the short rect-angles (b\/h < 0.683), showed amplitude responses when re-leased from rest which are s i m i l a r , and c h a r a c t e r i s t i c of a resonance phenomenon. The c i r c u l a r cylinder, which has p o s i t i v e aerodynamic damping f o r \u00b0C > 0\u00b0, showed a more l o c a l i z e d peak than the other two sections, but a l l three are found to vibrate within a li m i t e d speed range which includes the c r i t i c a l v e l o c i t y . However, the reversed D-section, the D-section i n i t i a l l y at 40\u00b0, and the long rect-angle (b\/h > 0.75) showed c h a r a c t e r i s t i c s which are not i n agreement with the vortex theory. The reversed D-section i s apparently quite stable f o r two reasons. F i r s t , i t has p o s i t i v e aerodynamic damping, and second, i t has no a f t e r -body extending into the wake. The D-section at P } 40\u00b0 i s also unaffected by vortex resonance, apparently because of very high p o s i t i v e aerodynamic damping. On the other hand, the long rectangles were found to vibrate at any airspeed above a certain minimum value. This behaviour i s charact-e r i s t i c of the galloping type of v i b r a t i o n and lends sup-port to Den Hartog's theory of i n s t a b i l i t y . I f Den Hartog's c r i t e r i o n i s applied to a l l the sections tested, the following r e s u l t i s obtained: a. D-Section B = 0\u00b0 Neutral i . e . dC F v\/d\u00b0< = 0 b. Reverse D-section Stable c. D-section with p = 40\u00b0 Stable d. C i r c u l a r cylinder Stable 58 e. Short rectangles (b\/h< 0.683) Neutral, dC F v\/cK = 0 f . Long rectangles (b\/h>0.75) Unstable Thus the theory i s v a l i d i n that none of the sections from \"a\" to \"e\" shows galloping behaviour when released from rest whereas \" f \" does. However, the theory does not explain the large amplitude o s c i l l a t i o n s obtained f o r \"a\", \"d\", and \"e\". The galloping theory has been applied by Parkin-s o n 1 1 to a square section cylinder with the same physical c h a r a c t e r i s t i c s as one of those tested. The r e s u l t s of h i s analysis are compared with the experimental amplitude response curves (Figs. 39, 53) with the time-amplitude curve? (F i g . 49) and with the time to maximum amplitude curves ( F i g . 50). The t h e o r e t i c a l r e s u l t s show good agree-ment with a l l the main features of the dynamic response of t h i s section. His analysis includes the e f f e c t of s t r u c t u r a l damping and gives good quantitative agreement with the s h i f t i n s t a r t i n g speed f o r increased damping. The slope of the asymptote to his curve was calculated and found to agree with the curve given by the energy bal-ance, i n d i c a t i n g that the polynomial approximations that he used f o r the aerodynamic c o e f f i c i e n t s were appropriate. The ordinates of his curve are also s l i g h t l y lower than the experimental values at the higher v e l o c i t i e s . This could be att r i b u t a b l e to the ef f e c t of vortex capture which was not accounted f o r i n h i s analysis. 59 The sections which showed lim i t e d i n s t a b i l i t y did so over a range of airspeeds which i n a l l cases includ-ed the c r i t i c a l v e l o c i t y V C p . I t therefore seems unlikely that the dynamic response of these sections could be pre-dicted by Steinman's theory since he does not consider the e f f e c t of vortex formation. I t appears necessary that any theory that w i l l account f o r aerodynamic exc i t a t i o n i n general w i l l have to use a combination of both negative aerodynamic damping and vortex e x c i t a t i o n . In the physical phenomenon, both effects are ob-served i n every case, During galloping, capture occurs, and the energy involved i n vortex formation becomes a v a i l -able to the v i b r a t i o n . S i m i l a r l y , the maximum amplitude that i s reached during a vortex type v i b r a t i o n i s l a r g e l y governed by the angle of attack at which the aerodynamic damping becomes strongly p o s i t i v e . Again, the l i f t and drag c o e f f i c i e n t s from which the damping force i s c a l c u l -ated are dependent on the wake pressure d i s t r i b u t i o n , which i n turn depends on the time average of vortex induced pressures. The i d e a l theory must be able to account f o r the two e f f e c t s separately, (as i n the case of the more heavily damped square section), i n combination near the c r i t i c a l v e l o c i t y , or a combination of capture and gallop-ing. I t must be able to predict the dynamic response of the following types of cross-sections: 60 Stable i . No afterbody and po s i t i v e aerodynamic damp-ing, i . e . , the reversed D-section. i i . Very high p o s i t i v e aerodynamic damping, i . e . the D-section i n i t i a l l y at 8 = 45\u00b0. Unstable over l i m i t e d speed range i . Small but pos i t i v e aerodynamic damping, i . e . the c i r c u l a r cylinder, i i . Zero aerodynamic damping for wide range of o< , i . e . , the D-section cylinder when re-leased from r e s t . Unstable i . Negative aerodynamic damping f o r small \u00b0< i . e . , the rectangles with b\/h } 0.75. i i . Zero aerodynamic damping for small \u00b0i , followed by a range of 0.75) developed an asymmetry at small o< which resulted i n nega-t i v e aerodynamic damping and galloping response. I t i s 63 f e l t that t h i s i s caused by the ef f e c t of the afterbody on the point of vortex formation which was found to be approx-imately 1,25 diameters downstream of the separation edges fo r a D-section. There i s a steady decrease i n amplitude with i n -creasing p f o r the square section at a given airspeed. There i s a corresponding decrease i n the angle at which the aerodynamic damping force becomes p o s i t i v e . I t was shown that the two variations are r e l a t e d . A s i m i l a r v a r i a t i o n was observed f o r the rectangular sections with increasing b\/h. Wake measurements behind a -series of s t a t i c rectangular models show a steady decrease i n S as b\/h increases. At b\/h = 2.5, there i s a sudden increase i n S. Observations i n the smoke tunnel have supported the suggest-ion that the sudden jump i s caused by the point of vortex formation s h i f t i n g from downstream of the t r a i l i n g edge, to a point between the t r a i l i n g and leading edges. Increasing the non-aerodynamic damping f o r a square section was found to increase the v e l o c i t y at which galloping started and also to separate the vortex resonance e f f e c t from the galloping. The eff e c t of the c r i t i c a l v e l -o c i t y on galloping was demonstrated by observation of the time required f o r the system to reach maximum amplitude at various airspeeds. For the high damping case the time re-quired near the speed at which the vi b r a t i o n s t a r t s i s very large. For the low damping case, the s t a r t i n g speed co-incides with the c r i t i c a l v e l o c i t y and the additional ex-c i t a t i o n due to vortex resonance r e s u l t s i n a much shorter time to huild-up. I t i s f e l t that f o r high v e l o c i t i e s , the e f f e c t s of the non-aerodynamic damping w i l l die out. In conclusion, i t i s recommended that some of the following t e s t s be performed: 1} Measurement of and f o r a square section cylinder with a s p l i t t e r plate i n the wake. The plate i n h i b i t s the formation of vortices close to the cylinder and the test would show whether or not the vortex induced pressures cause asym-metry of the wake pressure with . 2) Measurement of amplitude f o r a square section cy-l i n d e r over a f u l l range of 0. Comparison with amplitude calculated from an energy balance would serve as a check on the accuracy involved i n using the steady state c o e f f i c i e n t s i n the dynamic prob-lem. 3) Measurement of amplitude f o r a square section cy-l i n d e r with increased damping at high v e l o c i t i e s to observe whether or not the eff e c t of the damp-ing dies out. 4) Measurement of Cp>v at larger values of \u00b0< f o r the short rectangles to check f o r negative damp-ing. Also dynamic te s t s at higher v e l o c i t i e s and large i n i t i a l amplitudes to check f o r galloping i n s t a b i l i t y . 65 5) Measurement of spanwise co r r e l a t i o n of vortex formation f o r both stationary and vibrating models of a l l cross-sections. 6) Measurement of wake geometry and s t a b i l i t y while stationary and during v i b r a t i o n . Smoke tunnel tests indicate rapid d i f f u s i o n of the vortices downstream of the model for NR = 1500 to 8000. F i n a l l y , i t i s suggested that i n future dynamic t e s t s , small-er, l i g h t e r models should be used together with l i g h t e r springs i n order to reduce the v i b r a t i o n of the tunnel walls as much as possible. 66 APPENDIX A Use of the Lissajous E l l i p s e f o r the Determination of the Frequency of an Unknown Signal Let u=aCos(2irnt - q) be a signal of unknown f r e -quency n, and l e t v = bCos2irn ,t be a signal of known f r e -quency; Assume n = n 1, and l e t 2irnt = p Then u = aCos(p - q) (1) and v = bCosp (2) Expanding (1) u = a(CospCosq + SinpSinq) (3) Squaring (2) and (3) and adding, (u\/a) 2 + (v\/b) 2 = Cos 2pCos 2q + 2SinpSinq(CospCosq) + S i n 2 p S i n 2 q + Cos 2p Now, (2uvCosq)\/ab = 2Cos 2pCos 2q + 2SinpSinq(CospCosq) .*. (u\/a) 2 + (v\/b) 2 = Cos 2p - Gos 2pCos 2q + S i n 2 p S i n 2 q + (2uvCosq)\/ab = C o s 2 p ( l - Cos 2q) + Sin 2pSin 2q-+ (2uvCosq)\/ab = Sin 2q(Cos 2p + Sin 2p) + (2uvCosq)\/ ab = S i n 2 q + (2uvGosq)\/ab (u\/a) 2+ (v\/b) 2 (2uvCosq)\/ab - S i n 2 q = 0 (A) 67 Eqn.(4) i s the equations of an e l l i p s e whose major and minor axes depend on the amplitude of the signals u and v, and on the phase difference q. I f the signals u and v are put across the horizontal and v e r t i c a l input plates of an oscilloscope, the frequency of the signal u can be determined by varying the frequency of the known signal v, u n t i l a stable Lissajous e l l i p s e appears on the screen. The frequency of the known input w i l l then be exactly equal to that of the unknown s i g n a l . APPENDIX B 6 8 Wind Tunnel C o r r e c t i o n s There are only two wind tunnel c o r r e c t i o n s which are a p p l i c a b l e to the b l u f f c y l i n d e r s t e s t e d ; s o l i d b l o c k i n g and wake b l o c k i n g . S o l i d B l o c k i n g . \u00b0sb \" 0 . 3 3 3 A(irh\/2a) 2 where h = model width a = tunnel width A = a parameter which depends on a\/chord. Wake B l o c k i n g . \u00b0wb * 0.25cC D\/a where c \u00ab= chord of model I f a = o s b + a w b , then VQO = V T ( 1 + a) cLco = C L T ( 1 - 2a) cDco = CDT(1 \" 2o) The s u b s c r i p t 'T' denotes a wind t u n n e l measurement. MODEL ' CORRECTIONS 0 V V T CLoc\/ CLT CDco \/ CDT 2\" D-Section 0.0185 1.018 0.963 0.963 2\" C i r c u l a r Cylinder 0.0191 1.019 0.962 0.962 2 r t x l n Rectangle {c\/h = 2) 0.0229 1.023 0.955 0.955 2\"x2\" Rectangle (b\/h = 1) 0.0342 1.034 0.932 0.932 l \" x 2 \" Rectangle (b\/h = 0.5) 0.0185 1.018 0.963 0.963 APPENDIX C 7G The Den Hartog Instability Criterion Forces and angles are positive as shown Consider a body moving downward as shown with a velocity v, in the presence of a steady, uniform airstream of velocity V Q Q . Let V R be the relative velocity at an angle < to the horizontal. Then, V R = VQO - v (1) and, = Arctan(v\/Voo ) (2) The total upward damping force due to air pressure is given by, F y = DSin\u00b0< + LCos\u00b0< (3) If dFv\/d\u00ab< is negative, then the upward wind force, F , increases for negatively and decreases for positive VgD')Co.s2\"K' or, C F v = * CL + c DTan^ ) S e c o < APPENDIX E Aerodynamic Coef f i c i e n t s f o r the D-Section Cylinder Diameter = 2R By d e f i n i t i o n , C^ = L\/2R q o o C d = D\/2Rq o o P wRdo From the diagram, L = -2R pdsSin\u00abK+ 0 (ir\/2 + <) p^Sinada - V \/ 2 - <) 2R ( D = pdsCos\u00ab<-0 (ir\/2 + <<) pTrRCosodo -V\/2 - <) 2R C = - S i n * I pds + _ i 2Rqco 0 2qoo (ir\/2 + <<) p wSinado Air\/2 - c() and, 2R C d - Cos\u00b0< 2Rqoo . pds 2^00 0 (ir\/2 + \u00abK) p wCosodo -V\/2 - o C ) Now let p = (P + nKo)) , where P is a constant, Then, 2R (ir\/2-+ \u00b00 - Sin* I p d s + _P_ Sinodo 2 R W 0 2qco i(ir \/ 2 - \u00ab0 (ir\/2 + \u00ab0 n(%a)Sinodo 2 q \u00b0 \u00b0 - ) Now. (ir\/2 + * ) \u2022| Sinada = 2 S i n * _(ir\/2 - o< ) 2R (ir\/2 + \u2022<) - Sin* I pds + PSin\u00b0< + 1 I n(<*,a )Sinada 2R Let 1 2qaR pds - P\/q 00 ' 0 c l M and let n (\u00b0 ( ' q ? P ' 2qco - Z(%a) Then % -and Gd -(ir\/2 +<*) - Cp(*)Sin 8. 500' -0.50 -0.25 -0.0 0.25 0.50 (DISTANCE FROM MIDPOINT)\/h FIGURE 23 - PROJECTION OF C p ON UPSTREAM SURFACE (\u00ab 108 I. 00 \/ 0o50 0.00 B u u to co -0.50 -1,00 -1.50 1 y 1 i 5 * 2\" D -a = 15 N R = ct SECTION DEGREES J B U | W V ^ = 9.9\" o - X - 8.5\" -0.50 -0.25 0.0 o.25 i t (DISTANCE FROM MIDPOINT)\/h 0.50 FIGURE 24 - PROJECTION-OF C ON UPSTREAM SURFACE K=15\u00b0) P FIGURE 25 - MIDSPAN SECTIONAL COEFFICIENTS FOR D\u2014SECTION NR \u2022 66,000 110 1.00 0.50 cT o.oo w M o \u00a7 -0.50 o -1.00 -1*50 -2.00 2\" D - SECTION 2-DIMENSI0NAL a a 0\u00b0 15\u00b0 \u2014 A \u2014 A \u2014 FLAT PLATE a = 0\u00b0 REF. 4 -0.50 -0.25 0.0 0.25 0.50 (DISTANCE FROM MIDPOINT)\/h FIGURE 26 - PROJECTION OF C ON UPSTREAM SURFACE P TWO-DIMENSIONAL D-SECTION = 0\u00b0 & 15\u00b0 I l l -2o00 -0.50 -0.25 0.0 0.25 0.50 (DISTANCE FROM MIDPOINT)\/h FIGURE 27 - PROJECTION OF C ON UPSTREAM SURFACE TWO-DIMENSIONAL D-SECTION o( = 350 & 4 0 0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 V \/ V . n c ' c t FIGURE 32 AMPLITUDE AND VORTEX FREQUENCY OF D-SECTION - TORSIONAL MODE H H ON 3 3 - PLUNGING AND TORSIONAL AMPLITUDES FOR D-SECTION AT VARIOUS ANGLES OF ATTACK (p) FIGURE 36 - PLUNGING AND TORSIONAL MODES FOR CIRCULAR CYLINDER FIGURE LO - C ON 2:1 RECTANGLE AT VARIOUS a \u00a3 P FIGURE 46 - COMPARISON OF VELOCITY - AMPLITUDE CHARACTERISTICS 1.0 2o0 3.0 4.0 5.0 b\/h 6.0 FIGURE 48 - VARIATION OF WAKE WIDTH WITH DEPTH OF RECTANGLE FIGURE 50 - NUMBER' OF CYCLES TO MAXIMUM AMPLITUDE ( b \/ h = 1.0 ) FIGURE 51 DYNAMIC RESPONSE ~ GALLOPING F v(V,a) FIGURE 52 SPRING-MASS SYSTEM FOR ENERGY THEORY H NJ1 3.0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.6 8.0 V_ INCHES oo. FIGURE 55 - AMPLITUDE RESPONSE FOR SEVERAL SQUARE SECTION CYLINDERS FIGURE 57 - AMPLITUDE. RESPONSE OF D-SECTION SHOWING GALLOPING VIBRATION ","@language":"en"}],"Genre":[{"@value":"Thesis\/Dissertation","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0105916","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Program":[{"@value":"Mechanical Engineering","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"University of British Columbia","@language":"en"}],"Rights":[{"@value":"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","@language":"en"}],"ScholarlyLevel":[{"@value":"Graduate","@language":"en"}],"Title":[{"@value":"Experimental investigation of the aeroelastic instability of bluff two-dimensional cylinders","@language":"en"}],"Type":[{"@value":"Text","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/39542","@language":"en"}],"SortDate":[{"@value":"1960-12-31 AD","@language":"en"}],"@id":"doi:10.14288\/1.0105916"}